Geodesic
On the rank of random subsets of finite affine geometry
Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 209-217

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The aim of the paper is to give an effective formula for the calculation of the probability that a random subset of an affine geometry AG(r-1,q) has rank r. Tables for the probabilities are given for small ranks. The expected time to the first moment at which a random subset of an affine geometry achieves the rank r is derived.
Keywords: finite affine geometry, random matroids, hitting time 
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Kordecki, Wojciech. On the rank of random subsets of finite affine geometry. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 209-217. http://geodesic.mathdoc.fr/item/DMGT_2000_20_2_a4/